Note that §REF  implies this results for \(n\ge 2\) . For \(n=1\) , the result is known and follows from e.g. R. Bryant et al [1]}, [2]}. Note also that \((n+1=2)\) -dimensional projective structures admitting infinitesimal symmetries and the metrization problem for them was solved completely in [1]}, [4]}. As mentioned above, 2-dimensional projective structures are always variational. J. Lang in [5]} constructed Lagrangians for 2-dimensional path and projective structures with the submaximal symmetry algebra (of dimension 3), see also [6]}, [7]}.
